PROPERTIES OF A CLOSED POLYGON. 



411 



This work is independent of the path traversed, and is zero whenever 

 we return to the original level surface by making the mass describe 

 any given closed curve. Suppose that the two points P l and P 2 are 

 situate on the surface of the Earth, and that the mass is displaced 

 along this surface; the work of the vertical component is zero at 

 each instant, the expression for the work only depends on the 

 horizontal component H, and reduces to 



(2) 



v. 



fp. 



-v 2 = 

 J*l 



cose, 



the integral of the second member being zero whenever the mass is 

 made to describe a closed circuit. 



That being admitted, let us consider a polygon formed of great 

 circles passing through the points P , P I} P 2 (Fig. 94). Trace 



Fig. 94. 



at these various points the geographical meridians P M , P 1 M 1 , 

 P 9 M 2 , and the magnetic meridians P D , P^, P 2 D 2 . . . 



Let 



^o ^i> V-- be the declinations reckoned positively from north 

 to west; 



0.1 the azimuth of the arc P P 1 at the point P , this azimuth 

 being counted positively from north to east; 



1.0 the azimuth of the arc PQ?! at the point P x counted posi- 

 tively in the same direction ; 



c o-u i-o tf 16 values of the angles e at these various points. 



